9 Designs

9.6 Affine Planes

An affine plane A⁢G⁢(2,q) of order q is nothing but a 2−(q2,q,1) design i.e., 𝒫 consists of q2 points satisfying the following conditions:

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    Every line has q points.

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    Any two points lie on a unique line.

Now the design property guarantees us that any point lies on q+1 lines and there are q2+q lines. A point of difference with projective planes is that lines need not meet in affine planes i.e., parallel lines can exist.

Construction via projective planes:

Let (𝒫,ℒ) be a projective plane of order q and let L0∈ℒ be a fixed line. Consider the design with points 𝒫′=𝒫∖L0 and lines ℒ′={L∖L0:L∈ℒ,L≠L0}. The line L0 is called the line at infinity and for every L∈ℒ′, there exists a unique x∈L0 such that L∪{x0}∈ℒ. This is called the infinite point of L.

Exercise(A) 9.19.

Verify that ℒ′ is an affine plane of order q.

Construction from 𝔽q:

Let q be a prime power and set 𝒫=𝔽q×𝔽q. The lines are either of the form

L⁢(a,b)={(x,y)∈𝒫:y=a⁢x+b}

or

L⁢(c)={(c,y):y∈𝔽q},

where a,b,c∈𝔽q. Note that 𝒫 has cardinality q2 and every line has q points. We only need to show that any two points determine a line uniquely. Let p1=(x1,y1),p2=(x2,y2) be two distinct points. If x1=x2, then the points p1,p2 belong uniquely to the line L⁢(x1). If it were to belong to a line L⁢(a,b) then we have that y1−y2=a⁢(x1−x2)=0 implying y1=y2 and so violating the distinctness of the points. If x1≠x2, then the system of equations y1=a⁢x1+b,y2=a⁢x2+b has a unique solution a=y1−y2x1−x2, b=y2⁢x1−y1⁢x2x1−x2. So p1,p2∈L⁢(a,b) and hence showing that ℒ is an affine plane.

Exercise(A) 9.20.

Consider an affine plane of order q i.e., a 2−(q2,q,1) design. Let a parallel class be a set of mutually disjoint lines. Show the following.

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    Each parallel class contains q lines ;

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    there are q+1 such classes ;

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    any two lines from different classes intersect at a point ;

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    and lines of each parallel class form a partition of the points 𝒫.